Porwell temperaments

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This family of temperaments tempers out the porwell comma, [11 1 -3 -2⟩ = 6144/6125, and includes hendecatonic, hemischis, twothirdtonic, nessafof, septisuperfourth, whoops, and polypyth.

Discussed elsewhere are:

Hendecatonic

The hendecatonic temperament has a period of 1/11 octave, which represents 16/15 and four times of which represent 9/7.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 10976/10935

Mapping: [⟨11 0 43 -4], ⟨0 1 -1 2]]

mapping generators: ~16/15, ~3

Wedgie: ⟨⟨11 -11 22 -43 4 82]]

Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.054

Optimal ET sequence: 22, 55, 77, 99

Badness: 0.041081

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 10976/10935

Mapping: [⟨11 0 43 -4 38], ⟨0 1 -1 2 0]]

Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.636

Optimal ET sequence: 22, 55, 77, 99, 176e, 275e

Badness: 0.046088

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 351/350, 4459/4455

Mapping: [⟨11 0 43 -4 38 93], ⟨0 1 -1 2 0 -3]]

Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.291

Optimal ET sequence: 22, 55, 77, 99, 176e

Badness: 0.040099

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 176/175, 273/272, 2025/2023

Mapping: [⟨11 0 43 -4 38 93 45], ⟨0 1 -1 2 0 -3 0]]

Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.301

Optimal ET sequence: 22, 55, 77, 99, 176eg

Badness: 0.029054

Cohendecatonic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 4375/4356

Mapping: [⟨11 0 43 -4 73], ⟨0 1 -1 2 -2]]

Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.686

Optimal ET sequence: 22, 77e, 99e, 121, 220e

Badness: 0.038042

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 540/539, 625/624

Mapping: [⟨11 0 43 -4 73 128], ⟨0 1 -1 2 -2 -5]]

Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.888

Optimal ET sequence: 22, 77eff, 99ef, 121, 341bdeeff

Badness: 0.036112

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 364/363, 375/374, 540/539

Mapping: [⟨11 0 43 -4 73 128 45], ⟨0 1 -1 2 -2 -5 0]]

Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.877

Optimal ET sequence: 22, 77eff, 99ef, 121, 220efg, 341bdeeffgg

Badness: 0.022590

Icosidillic

Subgroup: 2.3.5.7.11

Comma list: 3388/3375, 6144/6125, 9801/9800

Mapping: [⟨22 0 86 -8 111], ⟨0 1 -1 2 -1]]

mapping generators: ~33/32, ~3

Optimal tuning (POTE): ~33/32 = 1\22, ~3/2 = 702.914

Optimal ET sequence: 22, 154, 176, 198

Badness: 0.057725

Twothirdtonic

Subgroup: 2.3.5.7

Comma list: 686/675, 6144/6125

Mapping: [⟨1 3 2 4], ⟨0 -13 3 -11]]

mapping generators: ~2, ~15/14

Wedgie: ⟨⟨13 -3 11 -35 -19 34]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 130.401

Optimal ET sequence: 9, 28b, 37, 46

Badness: 0.099601

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 686/675

Mapping: [⟨1 3 2 4 4], ⟨0 -13 3 -11 -5]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 130.430

Optimal ET sequence: 9, 28b, 37, 46

Badness: 0.040768

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 169/168, 176/175

Mapping: [⟨1 3 2 4 4 5], ⟨0 -13 3 -11 -5 -12]]

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 130.409

Optimal ET sequence: 9, 28b, 37, 46

Badness: 0.025941

Semaja

Subgroup: 2.3.5.7

Comma list: 3125/3087, 6144/6125

Mapping: [⟨1 -2 1 3], ⟨0 19 7 -1]]

mapping generators: ~2, ~8/7

Wedgie: ⟨⟨19 7 -1 -33 -55 -22]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4834

Optimal ET sequence: 16, 37, 53, 196d

Badness: 0.107023

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 3125/3087

Mapping: [⟨1 -2 1 3 1], ⟨0 19 7 -1 13]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4856

Optimal ET sequence: 16, 37, 53

Badness: 0.059838

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 176/175, 275/273

Mapping: [⟨1 -2 1 3 1 2], ⟨0 19 7 -1 13 9]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4794

Optimal ET sequence: 16, 37, 53

Badness: 0.032564

Nessafof

Cryptically named by Petr Pařízek in 2011[1], nessafof adds the landscape comma and has a third-octave period. The name actually refers to the fact that it has a neutral-second generator, and that a semi-augmented fourth, stacked 5 times, makes 5/1[2].

Subgroup: 2.3.5.7

Comma list: 6144/6125, 250047/250000

Mapping: [⟨3 2 5 10], ⟨0 7 5 -4]]

mapping generators: ~63/50, ~35/32

Wedgie: ⟨⟨21 15 -12 -25 -78 -70]]

Optimal tuning (POTE): ~63/50 = 1\3, ~35/32 = 157.480

Optimal ET sequence: 15, 54b, 69, 84, 99, 282, 381

Badness: 0.045048

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 250047/250000

Mapping: [⟨3 2 5 10 8], ⟨0 7 5 -4 6]]

Optimal tuning (POTE): ~63/50 = 1\3, ~12/11 = 157.520

Optimal ET sequence: 15, 54be, 69e, 84e, 99

Badness: 0.068427

Nessa

Subgroup: 2.3.5.7.11

Comma list: 441/440, 1344/1331, 4375/4356

Mapping: [⟨3 2 5 10 10], ⟨0 7 5 -4 1]]

Optimal tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.539

Optimal ET sequence: 15, 54b, 69, 84, 99e

Badness: 0.048836

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 364/363, 441/440, 625/624

Mapping: [⟨3 2 5 10 10 6], ⟨0 7 5 -4 1 13]]

Optimal tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.429

Optimal ET sequence: 15, 54bf, 69, 84, 99ef, 183ef, 282eeff

Badness: 0.037409

Aufo

For the 5-limit version of this temperament, see High badness temperaments #Untriton.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 177147/175616

Mapping: [⟨1 6 -7 19], ⟨0 -9 19 -33]]

mapping generators: ~2, ~45/32

Wedgie: ⟨⟨9 -19 33 -51 27 130]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.782

Optimal ET sequence: 53, 161, 214

Badness: 0.121428

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 177147/175616

Mapping: [⟨1 6 -7 19 1], ⟨0 -9 19 -33 5]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.811

Optimal ET sequence: 53, 108e, 161e

Badness: 0.088631

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 351/350, 58806/57967

Mapping: [⟨1 6 -7 19 1 -12], ⟨0 -9 19 -33 5 32]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.788

Optimal ET sequence: 53, 108e, 161e, 214ee

Badness: 0.058507

Aufic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5632/5625, 72171/71680

Mapping: [⟨1 6 -7 19 -25], ⟨0 -9 19 -33 58]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.800

Optimal ET sequence: 53, 108, 161, 214, 375

Badness: 0.075149

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 847/845, 4096/4095

Mapping: [⟨1 6 -7 19 -25 -12], ⟨0 -9 19 -33 58 32]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.796

Optimal ET sequence: 53, 108, 161, 214, 375, 589be

Badness: 0.039050

Whoops

For the 5-limit version of this temperament, see Very high accuracy temperaments #Whoosh.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 244140625/243045684

Mapping: [⟨1 17 14 -7], ⟨0 -33 -25 21]]

mapping generators: ~2, ~441/320

Wedgie: ⟨⟨33 25 -21 -37 -126 -119]]

Optimal tuning (POTE): ~2 = 1\1, ~441/320 = 560.519

Optimal ET sequence: 15, 122d, 137, 152, 608d, 623bd, 775bcd

Badness: 0.175840

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 6144/6125

Mapping: [⟨1 17 14 -7 10], ⟨0 -33 -25 21 -14]]

Optimal tuning (POTE): ~2 = 1\1, ~242/175 = 560.519

Optimal ET sequence: 15, 122d, 137, 152, 608de, 623bde, 775bcde

Badness: 0.043743

Polypyth

For the 5-limit version of this temperament, see High badness temperaments #Leapday.

Polypyth (46 & 121) tempers out the same 5-limit comma as the leapday temperament (29 & 46), but with the porwell (6144/6125) rather than the hemifamity (5120/5103) tempered out.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 179200/177147

Mapping: [⟨1 0 -31 52], ⟨0 1 21 -31]]

mapping generators: ~2, ~3

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.174

Optimal ET sequence: 46, 121, 167, 288b, 455bcd, 743bcd

Badness: 0.137995

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187, 6144/6125

Mapping: [⟨1 0 -31 52 59], ⟨0 1 21 -31 -35]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.177

Optimal ET sequence: 46, 121, 167, 288be, 455bcde

Badness: 0.051131

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 1716/1715

Mapping: [⟨1 0 -31 52 59 64], ⟨0 1 21 -31 -35 -38]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.168

Optimal ET sequence: 46, 121, 167, 288be

Badness: 0.030292

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 325/324, 352/351, 364/363, 1716/1715

Mapping: [⟨1 0 -31 52 59 64 39], ⟨0 1 21 -31 -35 -38 -22]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.168

Optimal ET sequence: 46, 121, 167, 288beg

Badness: 0.019051

Icositritonic

The icositritonic temperament (46 & 161) has a period of 1/23 octave, so six period represents 6/5 and nine period represents 21/16.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 9920232/9765625

Mapping: [⟨23 0 17 101], ⟨0 1 1 -1]]

mapping generators: ~1323/1280, ~3

Wedgie: ⟨⟨23 23 -23 -17 -101 -118]]

Optimal tuning (POTE): ~1323/1280 = 1\23, ~64/63 = 29.3586

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness: 0.196622

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 6144/6125, 35937/35840

Mapping: [⟨23 0 17 101 116], ⟨0 1 1 -1 -1]]

Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3980

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness: 0.064613

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 847/845, 3584/3575

Mapping: [⟨23 0 17 101 116 158], ⟨0 1 1 -1 -1 -2]]

Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.2830

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness: 0.040484

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 441/440, 561/560, 847/845, 1089/1088

Mapping: [⟨23 0 17 101 116 158 94], ⟨0 1 1 -1 -1 -2 0]]

Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.2800

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness: 0.024676

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 441/440, 456/455, 476/475, 513/512, 847/845

Mapping: [⟨23 0 17 101 116 158 94 207], ⟨0 1 1 -1 -1 -2 0 -3]]

Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3760

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness: 0.021579

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 276/275, 351/350, 391/390, 441/440, 456/455, 476/475, 847/845

Mapping: [⟨23 0 17 101 116 158 94 207 104], ⟨0 1 1 -1 -1 -2 0 -3 0]]

Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3471

Optimal ET sequence: 46, 115, 161, 207, 368ci

Badness: 0.017745

Countermiracle

The countermiracle temperament (31 & 145) tempers out the trimyna, 50421/50000 and the quince comma, 823543/819200.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 50421/50000

Mapping: [⟨1 4 3 3], ⟨0 -25 -7 -2]]

mapping generators: ~2, ~343/320

Wedgie: ⟨⟨25 7 2 -47 -67 -15]]

Optimal tuning (POTE): ~2 = 1\1, ~343/320 = 115.9169

Optimal ET sequence: 31, 114, 145, 176, 559cc, 735cc

Badness: 0.102326

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 6144/6125

Mapping: [⟨1 4 3 3 8], ⟨0 -25 -7 -2 -47]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9158

Optimal ET sequence: 31, 114e, 145, 176

Badness: 0.039162

Countermiraculous

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 1001/1000, 6144/6125

Mapping: [⟨1 4 3 3 8 1], ⟨0 -25 -7 -2 -47 28]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8803

Optimal ET sequence: 31, 83e, 114e, 145, 321ceff

Badness: 0.039271

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224

Mapping: [⟨1 4 3 3 8 1 1], ⟨0 -25 -7 -2 -47 28 32]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8756

Optimal ET sequence: 31, 83e, 114e, 145

Badness: 0.029496

Counterbenediction

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 3146/3125, 3584/3575

Mapping: [⟨1 4 3 3 8 -2], ⟨0 -25 -7 -2 -47 59]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9335

Optimal ET sequence: 31, 114ef, 145f, 176, 207, 383c, 590cc

Badness: 0.045569

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125

Mapping: [⟨1 4 3 3 8 -2 -2], ⟨0 -25 -7 -2 -47 59 63]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9391

Optimal ET sequence: 31, 114efg, 145fg, 176, 207

Badness: 0.036289

Countermanna

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 3388/3375, 6144/6125

Mapping: [⟨1 4 3 3 8 15 0 -25 -7 -2 -47 -117]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8898

Optimal ET sequence: 145, 176, 321ce

Badness: 0.053409

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375

Mapping: [⟨1 4 3 3 8 15 15], ⟨0 -25 -7 -2 -47 -117 -113]]

Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8832

Optimal ET sequence: 145, 321ce

Badness: 0.040898

Counterrevelation

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 50421/50000

Mapping: [⟨1 4 3 3 5], ⟨0 -25 -7 -2 -16]]

Optimal tuning (POTE): ~2 = 1\1, ~343/320 = 115.9192

Optimal ET sequence: 31, 114, 145e, 176e

Badness: 0.064070

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 196/195, 13750/13689

Mapping: [⟨1 4 3 3 5 1], ⟨0 -25 -7 -2 -16 28]]

Optimal tuning (POTE): ~2 = 1\1, ~273/256 = 115.8624

Optimal ET sequence: 31, 83, 114, 145e

Badness: 0.057497

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625

Mapping: [⟨1 4 3 3 5 1 1], ⟨0 -25 -7 -2 -16 28 32]]

Optimal tuning (POTE): ~2 = 1\1, ~91/85 = 115.8527

Optimal ET sequence: 31, 83, 114, 145e

Badness: 0.044043

Absurdity

For the 5-limit version of this temperament, see High badness temperaments #Absurdity.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 177147/175000

Mapping: [⟨7 0 -17 64], ⟨0 1 3 -4]]

mapping generators: ~972/875, ~3

Optimal tuning (POTE): ~972/875 = 1\7, ~3/2 = 700.5854 (or ~10/9 = 186.2997)

Optimal ET sequence: 77, 84, 161

Badness: 0.133520

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 6144/6125, 72171/71680

Mapping: [⟨7 0 -17 64 124], ⟨0 1 3 -4 -9]]

Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 700.6354 (or ~10/9 = 186.3497)

Optimal ET sequence: 77, 84, 161

Badness: 0.081564

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 1188/1183, 3584/3575

Mapping: [⟨7 0 -17 64 124 37], ⟨0 1 3 -4 -9 -1]]

Optimal tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6291 (or ~10/9 = 186.3434)

Optimal ET sequence: 77, 84, 161

Badness: 0.041600

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625

Mapping: [⟨7 0 -17 64 124 37 -49], ⟨0 1 3 -4 -9 -1 7]]

Optimal tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6524 (or ~10/9 = 186.3667)

Optimal ET sequence: 77, 161

Badness: 0.031783

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494

Mapping: [⟨7 0 -17 64 124 37 -49 63], ⟨0 1 3 -4 -9 -1 7 -3]]

Optimal tuning (POTE): ~21/19 = 1\7, ~3/2 = 700.6565 (or ~10/9 = 186.3708)

Optimal ET sequence: 77, 161

Badness: 0.022291

Dodifo

For the 5-limit version of this temperament, see High badness temperaments #Dodifo.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 2500000/2470629

Mapping: [⟨1 12 5 4], ⟨0 -35 -9 -4]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 357.070

Optimal ET sequence: 37, 84, 121, 205

Badness: 0.179692

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 2560/2541, 4375/4356

Mapping: [⟨1 12 5 4 -1], ⟨0 -35 -9 -4 15]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 357.048

Optimal ET sequence: 37, 84, 121, 326dee

Badness: 0.081923

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 625/624, 640/637, 1375/1372

Mapping: [⟨1 12 5 4 -1 4], ⟨0 -35 -9 -4 15 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 357.049

Optimal ET sequence: 37, 84, 121, 326deef

Badness: 0.039533

  1. ↑ Yahoo! Tuning Group | Some more unclassified temperaments
  2. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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